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Question Number 177933 by Spillover last updated on 11/Oct/22

$$\mathrm{If}1+\sin \mathrm{x}+\sin {}^2\mathrm{x}+\sin {}^3\mathrm{x}+...\mathrm{\infty }$$$$=4+2\sqrt{3},0<\mathrm{x}<\pi \mathrm{Find}\mathrm{the}\mathrm{value}\mathrm{of}\mathrm{x}$$$$$$

Answered by Ar Brandon last updated on 11/Oct/22

$$1+\sin x+{\sin }^{2}x+{\sin }^{3}x+\cdot \cdot \cdot =\frac{1}{1-\sin x}=4+2\sqrt{3}$$$$\Rightarrow \frac{1}{1-\sin x}=2(2+\sqrt{3})=\frac{2}{2-\sqrt{3}}=\frac{1}{1-\frac{\sqrt{3}}{2}}$$$$\Rightarrow \sin x=\frac{\sqrt{3}}{2}\Rightarrow x=\frac{\pi }{3},\frac{2\pi }{3}$$

Commented by Spillover last updated on 11/Oct/22

$$thanksforsolving$$

Answered by Rasheed.Sindhi last updated on 11/Oct/22

$$1+\sin \mathrm{x}+\sin {}^2\mathrm{x}+\sin {}^3\mathrm{x}+...\mathrm{\infty }=4+2\sqrt{3}$$$$\sin \mathrm{x}+\sin {}^2\mathrm{x}+\sin {}^3\mathrm{x}+...\mathrm{\infty }=3+2\sqrt{3}$$$$\sin \mathrm{x}(1+\sin \mathrm{x}+\sin {}^2\mathrm{x}+\sin {}^3\mathrm{x}+...\mathrm{\infty })=3+2\sqrt{3}$$$$\sin \mathrm{x}(4+2\sqrt{3})=3+2\sqrt{3}$$$$\sin \mathrm{x}=\frac{3+2\sqrt{3}}{4+2\sqrt{3}}\cdot \frac{4-2\sqrt{3}}{4-2\sqrt{3}}=\frac{12-4\left(3\right)+8\sqrt{3}-6\sqrt{3}}{16-4\left(3\right)}$$$$\sin \mathrm{x}=\frac{2\sqrt{3}}{4}=\frac{\sqrt{3}}{2}\Rightarrow \mathrm{x}=60°,120°$$

Commented by Spillover last updated on 11/Oct/22

$$\mathrm{nice}\mathrm{solution}$$

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